June 27, 2008

Ian Stewart traced the history of the mathematics of four or more dimensions as it evolved from an an obscure and esoteric collection of ideas during the Victorian period into something “straightforward and ubiquitous.” Victorian spiritualists attempted to use emerging concepts of the fourth dimension to explain their esoteric beliefs using mathematical logic. This is echoed today in a myriad of print and virtual publications and postings where quasi-scientific explanations are offered by well-meaning people and charlatans alike to promote their views on spirituality using “mathematics” and “physics.” Rather than strengthening their own spiritual orientations or religions, they undermine more profound philosophical arguments by deflecting uplifting conversations into the realm of science fiction and transforming potential conversations on spirituality into arguments about fractals, numerology, quantum physics, etc.

“Why is it so hard to apply reasoning with the aspirational symmetry of mathematics to the other aspects of our lives?”

It is indeed more difficult to apply reasoning since the process of tracing sources of knowledge claims has become even more time-intense with the proliferation of information. In order to reason and use logic we need to start with some truth claims we can trust. We live in a period of democratization of knowledge management. Whose information do we trust?

“We do not see how things are. We see things as we are.” Talmud

In contrast, Ian Stewart (1945-), Fellow of the Royal Society (2001-), professor of mathematics at Warwick University, UK, and a science and science fiction writer as well as Director of the University of Warwick’s Mathematics Awareness Centre has popularized mathematics by making it more accessible. In his most recent publication is entitled Why Truth is Beauty: A History of Symmetry, he writes eloquently about the resonance between mathematics and the natural world:

“Throughout history, mathematics has been enriched from two different sources. One is the natural world, the other the abstract world of logical thought. It is these two in combination that give mathematics its power to inform us about the universe…The story of symmetry demonstrates how even a negative answer to a good question can lead to deep and fundamental mathematics . . . The true strength of mathematics lies precisely in this remarkable fusion of the human sense of pattern (Beauty) with the physical world, which acts both as a reality check (Truth) and as an inexhaustible source of inspiration (Stewart 2007 WTB: HS).”

Webliography and Bibliography

Abbott Abbott, Edwin. 1884. Flatland.

The Victorian clergyman, schoolmaster, and Shakespearean scholar created the character Hex, a small talking orange heretic hexagon living in a (flatland) Euclidean plane, granddaughter of Arthur Square. Hex pondered on the possibility that, “[A] square … moving somehow parallel to itself … can make something else, [a] supersquare that represents three to the third power, or 27 units; [A] supersquare in the third dimension!”. In an elegant, old-fashioned style Abbott Abbott described the “dimensional analogy” between the “Flatland view of 2-space and the human view of 3-space with some biting social criticism of the Victorian treatment of two disadvantaged groups: women and the poor.” See Stewart (2007-11).

Banchoff, Tom. geometer and Flatland authority,

Dewdney, Kee. 1984. The Planiverse.

Hinton, Charles Howard. 1907. An Episode of Flatland.

Stewart, Ian. 2007a. Why Truth is Beauty: A History of Symmetry. Basic Books

“Hidden in the heart of the theory of relativity, quantum mechanics, string theory, and modern cosmology lies one concept: symmetry. Symmetry has been a key idea for artists, architects and musicians for centuries, but within mathematics it remained, until recently, an arcane pursuit. In the twentieth century, however, symmetry emerged as central to the most fundamental ideas in physics and cosmology. Why Truth is Beauty: A History of Symmetry tells its history, from ancient Babylon to twenty-first century physics. It is a peculiar history, and the mathematicians who contributed to symmetry’s ascendancy mirror its fascinating puzzles and dramatic depth. We meet Girolamo Cardano, the Renaissance Italian rogue, scholar, and gambler who stole the modern method of solving cubic equations and published it in the first important book on algebra. We meet Evariste Galois, a young revolutionary who single handedly refashioned the whole of mathematics by founding the field of group theory—only to die at age nineteen in a duel over a woman before publishing any of his work. Perhaps most curious is William Rowan Hamilton, who carved his most significant discovery into a stone bridge between bout of alcoholic delirium (cover).”

“The late Victorian era, in short, was a period of remarkable progress and free thinking. Science and the Church came to a gentleman’s agreement not to tread on each others’ toes, and many a country clergyman became the world expert on six types of beetle or the reproductive habits of slugs. Scientific advances were discussed along with the price of sugar and the increasingly parlous state of the British Empire at garden parties and polite social gatherings. Victorians, in particular, were fascinated by “the” fourth dimension. Mathematicians had come to recognise that the dimensionality of our own familiar space did not necessarily impose constraints on the dimensionality of any other structure. Mathematics was littered with “spaces” of dimension four, or ten, or a hundred. Many of these spaces accurately represented aspects of the physical world—for instance, as “degrees of freedom” of a mechanical system. Spiritualism, another flourishing Victorian interest, latched on to the fourth dimension as a convenient location for the spirit world. Ghosts could enter our world “sideways” along a dimension that mere humans could not observe or experience. Hyperspace theologians seized on the fourth dimension as an excellent place in which to put God and His angels, though they quickly realised that the fifth, sixth, and seventh dimensions were even better, and the infinitieth dimension added a satisfactory element of closure. While well-meaning people and charlatans of every kind were appealing to the fourth dimension to justify their beliefs and scams, the mathematics of four or more dimensions was changing from an obscure and esoteric collection of ideas into something straightforward and ubiquitous (Stewart 2007-11:1318-9).”

“[At] the dawn of the New Millennium, Arthur receives a visit from Spherius the sphere, an extradimensional being from the mysterious world of Space. But this is no dream. A quick run through the dimensional analogy fails to convince Arthur that Space can possibly exist, so Spherius bumps him out of Flatland. Now he sees the entire plane spread out before him. “I can see inside everything … I can see inside everyone’s bodies and I’m going to be sick.” He even sees the artifact in Area 33H, which is a slowly spinning cube. Now Arthur realises that Hex was right—a supersquare does exist. And back to Flatland he goes, to spread the gospel of the Third Dimension.” (Stewart 2007-11:1321).